The Hirshhorn Museum’s board deadlocked Thursday on whether to forge ahead with the avant-garde inflatable structure known as the Bubble, and the museum’s director, Richard Koshalek, announced shortly afterward that he’ll quit later this year. Koshalek proposed the ambitious project four years ago to elevate the Hirshhorn’s international profile, seeking a bold commitment to architectural innovation and a new program of arts and cultural events.

The approximately 150-foot-tall seasonal inflatable structure, which would cost an estimated $15.5 million, would be inflated during clement months for special events, symposia and programs supporting the Hirshhorn’s exhibition schedule.

Thursday’s vote made it clear that the Smithsonian is at a cultural crossroads, and Clough’s decision will do more than decide the fate of what is officially called the Seasonal Inflatable Structure. It will also pass judgment on Koshalek’s legacy and vision, which was to make the Hirshhorn a more open, engaged and internationally important institution. If Clough kills the project, grave doubts would be raised about whether the Smithsonian can attract a top-tier successor to Koshalek, and whether it is a fit steward for a world-class art museum.

Sending the final decision to Clough, however, underscores the inherent tension between the Hirshhorn and its parent institution. Under Koshalek’s guidance, the museum was positioning itself as daring and risk-taking — qualities not in abundance at the Smithsonian Castle, which manages a portfolio of more traditional museums and cultural and scientific institutions, and which is traditionally hypersensitive to controversy and conflict. If Clough votes down the Bubble, it will send a clear and chilling message to other Smithsonian entities and to anyone who is courted to replace Koshalek.

Even though he announced he will be leaving the Hirshhorn, Koshalek was still strongly supportive of the Bubble idea on Thursday.

“There are very few moments when an institution — even one as great as the Smithsonian — can take a global leadership role in the larger worlds of education and contemporary information exchange,” he said. “You don’t let this opportunity pass you by.”

Clough has earned a reputation atop the Smithsonian for being risk-averse, most notably in late 2010, when he personally intervened to remove material from an exhibition about gay and lesbian portraiture, “Hide/Seek,’’ at the Smithsonian’s National Portrait Gallery. Overruling respected curators and the museum’s leadership, Clough bowed to political pressure from a handful of conservative legislators and Catholic groups and censored a historic video — a work by David Wojnarowicz — that was deemed by some critics to be offensive to religious sensibilities.

The decision sparked protests from artists and museum professionals around the world, and it raised questions about whether Clough, an engineer who came to the institution from the Georgia Institute of Technology, understood the cultural and artistic side of the Smithsonian’s mission.

None of this bodes well for how he will wield his power in the case of the Hirshhorn Bubble. Sources close to the deliberations say that while the Smithsonian has already supported the Bubble with $4 million of funding, top leadership isn’t behind the project, despite professing enthusiasm for the concept. The leak last week of a slapdash report on the possible costs of supporting programming at the Bubble suggests that important figures within the Hirshhorn or the Smithsonian were working to influence the outcome of the Hirshhorn board vote.

Although critics of the Bubble have cited its cost during a period of lean budgets as a major reason not to go forward, Clough’s decision could put the Hirshhorn at even greater peril. Hanging over this entire debate is the specter of the Corcoran Museum, which hired architect Frank Gehry to design an addition to the museum, then voted in 2005 not to proceed with the project because of fundraising concerns. David C. Levy, the director of the Corcoran at the time, resigned immediately after that decision, and the museum and art school have been plagued since then by even deeper fundraising woes, plus confusion about its mission and priorities.

The Corcoran was and is a far more troubled institution than the Hirshhorn, which is supported by government funding. But the crisis precipitated by the Corcoran’s decision to scuttle the Gehry project illustrates the speculative nature of art and art museums: Institutions grow by announcing ambitious plans and building support for them, but there is a powerful risk that by failing to realize a bold plan, support withers and momentum dies. If the Bubble is scrapped, anyone who considers replacing Koshalek will naturally wonder whether the Smithsonian is capable of hosting a forward-thinking art museum under its institutional umbrella.

The brilliance of the Bubble idea, as designed by the New York-based architecture firm of Diller Scofidio + Renfro, is that it would cost less than one-tenth of the projected cost of the Gehry wing at the Corcoran, yet would have an outsized impact on architectural thinking in Washington. Koshalek and his architects have finessed one of this city’s hardest design challenges: How to make something new, in the center of the city, but in such a way that no permanent violence is done to the historic character of the Mall and its environs.

The Bubble would also bring innovative temporary architecture — one of the most intellectually exciting currents in contemporary architectural thinking — to the District. It would enliven the generally moribund civic space of the Mall, and demonstrate something that is now seriously in question: that Washington has developed progressive instincts when it comes to architecture, design and culture.

Critics of Koshalek have argued that he is directing institutional energies away from the Hirshhorn’s primary focus: the acquisition, display, study and preservation of modern and contemporary art. The Bubble, they argue, is a distraction. Koshalek certainly did bring a sense of showmanship and West Coast energy to Washington cultural life, but it has been in service to building an audience for contemporary art. With projects such as last year’s “Song 1,’’ which projected a video work by artist Doug Aitken onto the exterior wall of the museum, Koshalek proved he could break down the museum’s isolation, so perfectly expressed in its architectural form: An almost windowless concrete cylinder.

Since he arrived in 2009, Koshalek has rigorously addressed the most obvious and fundamental question of any art museum today: How to break out of the institutional bunker, and into a larger dialogue with the city and the country at large?

The Bubble was his answer. If Clough vetoes it, the Hirshhorn will be back to square one, forced to reconsider the question Koshalek already answered. Or else it will retreat into itself and succumb to isolation, repetition and complacency.

The decision is important enough that it shouldn’t be left in the hands of Clough. This is the time for the regents of the Smithsonian to engage with the project and make clear their vision of the institution. Is there room for an art museum within the institutional culture of the Smithsonian? Does the Smithsonian want to participate in defining the role of culture in American life?

Voting down the Bubble would effectively say no to all those questions, and the impact on innovation and ideas at the Smithsonian wouldn’t just be dispiriting, it would — to paraphrase the Smithsonian’s marketing — be seriously amazing.

Philip Kennicott is the Pulitzer Prize-winning Art and Architecture Critic of The Washington Post. He has been on staff at the Post since 1999, first as Classical Music Critic, then as Culture Critic.

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